In this paper the techniques used by Kac and Siegert and by Emerson for evaluating the probability distribution for the filtered output of a square-law device with a stationary, Gaussian input, have been extended to the case of a multiplier whose inputs are a pair of correlated, stationary, Gaussian time-series. It is shown that in this case the probability distribution is determined by the eigenvalues of a pair of simultaneous, linear, homogeneous, integral equations whose kernels involve only the correlation functions of the inputs and the impulse response of the postmultiplier filter. Explicit solutions for the eigenvalues of these integral equations are obtained both for the case of no postmultiplier filtering and for a simple example system using RC filters. Using these solutions the corresponding probability distributions are discussed and in particular, the way in which the probability distribution of the output tends to Gaussian as the postmultiplier filter time constant is increased, is demonstrated.

In this paper it is shown that, in general, second-order probability distributions may be expanded in a certain double series involving orthogonal polynomials associated with the corresponding first-order probability distributions. Attention is restricted to those second-order probability distributions which lead to a "diagonal" form for this expansion. When such distributions are joint probability distributions for samples taken from a pair of time series, some interesting results can be demonstrated. For example, it is shown that if one of the time series undergoes an amplitude distortion in a time-varying "instantaneous" nonlinear device, the covariance function after distortion is simply proportional to that before distortion. Some simple results concerning conditional expectations are given and an extension of a theorem, due to Doob, on stationary Markov processes is presented. The relation between the "diagonal" expansion used in this paper and the Mercer expansion of the kernel of a certain linear homogeneous integral equation, is pointed out and in conclusion explicit expansions are given for three specific examples.